By Charles B. Officer

ISBN-10: 3642657311

ISBN-13: 9783642657313

ISBN-10: 3642657338

ISBN-13: 9783642657337

It has been my purpose during this publication to provide a coordinated remedy of the total of theoretical geophysics. The ebook assumes a mathematical again floor via calculus and differential equations. It additionally assumes a cause capable historical past in physics and in basic vector research. the extent of the publication is commensurate with that of a senior undergraduate or first yr graduate path. Its target is to supply the reader with a survey of the complete of theoretical geophysics. The emphasis has been at the uncomplicated and the ordinary. The specialist in anyone of different disciplines lined the following will locate a lot missing from his specific sector of research; no apology is made for that. so one can deal with all elements in a coordinated demeanour, the easiest kind of mathematical nota tion for many of the actual difficulties has been used, specifically, that of scalars, 3-dimensional vectors, and the vector operators, gradient, curl, divergence, and so on. it really is preferred that this ordinary notation frequently is probably not the main conducive to the answer of a few of the extra complicated geophysical difficulties. The derivations are, in nearly each case, carried via in significant element. occasionally the details of the algebra and calculus were passed over and relegated to at least one of the issues following the part. The emphasis has been at the physics of the derivations and on explaining a few of the actual ideas vital in geophysics, resembling continuity, blending, diffusion, conduction, convection, precession, wobble, rays, waves, dispersion, and strength theory.

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**Example text**

L67) The Fourier transforms, either Eqs. l65) or more generally Eqs. 167), have a useful application in the description of the source function for problems in vibration or wave propagation. Often, we wish to consider a source functionf(t), which is usually given as some function of the pressure, velocity, or displacement with respect to time in terms of its simple harmonic components. By a simple harmonic component, we mean an oscillation with respect to time of a cosine or sine expression. 168) Its graph will be as shown in Fig.

We now wish to derive certain recursion formulas among the Pn and their derivatives. Taking the derivative of Eq. 95) Comparing Eq. 95) with Eq. 93), we may write, rearranging terms, (u-x) [PO+P I +P2X2 + ... +Pn~+···J = (1-2ux+x 2 ) [PI +2Pz x+3P3 X2+ ... 96) Now taking the derivative of Eq. 93) with respect to u, we obtain x 23/2 = (1-2ux+x) P~(u)+P{(u)x+p~(U)X2+... 97) Again comparing this time Eq. 97) with Eq. 95), we may write, rearranging terms, (u-x) [P~+P{X+p~X2+ ... +P~~+···J = x[P 1+2P2x+3P3 X2 + ...

210) K. Carry through the reductions to obtain Eqs. 11 Associated Legendre Polynomials Although we shall be concerned almost entirely, at the level of this book, with azimuthal symmetry for those problems utilizing spherical coordinates, it is of interest to develop the functions that occur in the three-dimensional solution of Laplace's equation in spherical coordinates. Such solutions involving these functions necessarily occur in many geophysical problems involving solutions of Laplace's equation for gravitational, hydrodynamic, and magnetic potentials.

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